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(Bistable Simulation Without "Switch")
(Bistable Simulation Without "Switch")
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===Bistable Simulation Without "Switch"===
===Bistable Simulation Without "Switch"===
<math>\frac{d[A]}{dt} = \frac{K_{1}[A]}{K+c[A]+c'[iA]}-K_{d}[A]</math>
<math>\frac{d[A]}{dt} = \frac{K_{1}[A]}{K+c[A]+c'[iA]}-K_{d}[A]</math>

Revision as of 17:44, 27 October 2012

DNA tablet



Bistable System

Using the bistable system, you can realize two different states, state"A" and state"B". State"A" has many staple strands of type A and few of type B. In contrast, "B" has few staple strands of type A and many of type B.

Please note that we'll represent state"X" as "X", and staple strands of type X as X.

How do concentrations of A and B change? In this section, we present the results of the simulations.

Bistable Simulation Without "Switch"


\frac{d[A]}{dt} = \frac{K_{1}[A]}{K+c[A]+c'[iA]}-K_{d}[A]

\frac{d[B]}{dt} = \frac{K_{1}[B]}{K+c[B]+c'[iB]}-K_{d}[B]

\frac{d[iA]}{dt} = \frac{K_{2}[B]}{K'+[B]}-K_{d}[iA]

\frac{d[iB]}{dt} = \frac{K_{2}[A]}{K'+[A]}-K_{d}[iB]

K_{1}=K_{2}=1 \quad K_{d}=0.1 \quad K=K'=1 \quad c=1 \quad c'=2

[A]_{0}=3.0 \quad [B]_{0}=2.9 \quad [iA]_{0}=0 \quad [iB]_{0}=0

As you can see, even with only a small difference between the initial concentrations [A] and [B], the appropriate state is reached.

How to Set up Equations

The picture below shows all of the reactions. However, as we'll explain later, this is a simplified model. For an introduction about these reactions, please refer to Idea.

A\xrightarrow{T_{A}}2A \quad A\xrightarrow{T_{iA}}A+iB

B\xrightarrow{T_{B}}2B \quad B\xrightarrow{T_{iB}}B+iA

A \to 0 \quad iA \to 0

B \to 0 \quad iB \to 0

For example, left of the top reaction formula shows that A is doubled by template TA. Please note that A, B, iA, and iB naturally decrease because exonuclease gradually decomposes these staple strands.

To set up the equations, let's have a closer look at some reactions.

First of all, let's consider the reactions among A, iA and TA

TA has three states.

TA  :normal state
ATA  :TA hybridizing with A
iATA  :TA hybridizing with iA

The sum is constant.

TA + ATA + iATA = n

We considered A increases in proportion to \frac{AT_{A}}{n}

A+T_{A} \rightleftarrows AT_{A} \to 2A+T_{A}

iA+T_{A} \rightleftarrows iAT_{A}

[TA] + [ATA] + [iTA] = n

\frac{[A][T_{A}]}{[AT_{A}]}=L_{1} \quad \frac{[iA][T_{A}]}{[iAT_{A}]}=L_{2}

The conclusion below is derived from these equations.


\frac{d[A]}{dt} = \frac{K_{1}[A]}{K+c[A]+c'[iA]}-K_{d}[A]

This is the equation we showed at the top of this subsection. The other equations can be obtained the same way.

Bistable Simulation with "Switch"

The bistable system magnifies even small differences. If you want to change the state from "A" to "B", all you have to do is to add more B than A when at the equilibrium. But, in this case, there are also many iB, so you have to add excessive amount of B.

This graph shows such operations at t=300 and t=600.

The Change between Two Pictures

Next simulation is about the process of changing the surface of the DNA tablet between the two states; "A" and "B". Let's note a hybridized strand as "1", and a non-hybridized strand strand as "0". On the surface of the tablet, two types of reaction occur; 1→0 and 0→1. Because denaturation occurs at a constant rate, the event probability of "1 to 0" does not change, while that of "0 to 1", i.e. hybridization, is proportional to the concentration of the complementary strands existing around the tablet.

Two simulations below: one is very simple, with a small tablet, only 5×5 pixels, and the other is with a larger one, 9×12. In the simulation movies, the blue points represent the hybridized strands "1". In these simulation, the concentration curves of every element are the same to those of the bistable simulation (look at the graph in the last section).

5×5 Origami Simulation

state A : "A" appears. state B : "B" appears.

9x12 Origami Simulation (the actual design of the DNA tablet)


This is the simulation of the DNA tablet with the actual design. One picture is "Tablet Boy" and the other is "I love DNA".

DNA Tablet with N-stable System and N-oscillator System

N-stable System Simulation

Tristable system:

Quintistable system:

N-oscillator System Simulation

Trioscillator system:

Quintioscillator system:

And a simulation video:

In this short movie, the concentration change of the simulation above is applied. The state changes from "only A" to "B" to "C" to "A" ..., and the vertical line moves from left to right.

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